library(tidyverse) #for data wrangling etc
library(rstanarm) #for fitting models in STAN
library(cmdstanr) #for cmdstan
library(brms) #for fitting models in STAN
library(car)
library(tidybayes) #for more tidying outputs
library(standist) #for exploring distributions
library(coda) #for diagnostics
library(bayesplot) #for diagnostics
library(ggmcmc) #for MCMC diagnostics
library(rstan) #for interfacing with STAN
library(HDInterval) #for HPD intervals
library(emmeans) #for marginal means etc
library(DHARMa) #for residual diagnostics
library(broom) #for tidying outputs
library(ggeffects) #for partial plots
library(broom.mixed)#for summarising models
library(ggeffects) #for partial effects plots
library(patchwork) #for multiple plots
library(ggridges) #for ridge plots
library(bayestestR) #for ROPE
library(see) #for some plots
library(easystats) #for the easystats ecosystem
source("helperFunctions.R")Bayesian GLM Part4
1 Preparations
Load the necessary libraries
2 Scenario
Loyn (1987) modelled the abundance of forest birds with six predictor variables (patch area, distance to nearest patch, distance to nearest larger patch, grazing intensity, altitude and years since the patch had been isolated).
| ABUND | DIST | LDIST | AREA | GRAZE | ALT | YR.ISOL |
|---|---|---|---|---|---|---|
| .. | .. | .. | .. | .. | .. | .. |
| ABUND | Abundance of forest birds in patch- response variable |
| DIST | Distance to nearest patch - predictor variable |
| LDIST | Distance to nearest larger patch - predictor variable |
| AREA | Size of the patch - predictor variable |
| GRAZE | Grazing intensity (1 to 5, representing light to heavy) - predictor variable |
| ALT | Altitude - predictor variable |
| YR.ISOL | Number of years since the patch was isolated - predictor variable |
The aim of the analysis is to investigate the effects of a range of predictors on the abundance of forest birds.
3 Read in the data
Rows: 56 Columns: 7
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (7): ABUND, AREA, YR.ISOL, DIST, LDIST, GRAZE, ALT
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.Rows: 56
Columns: 7
$ ABUND <dbl> 5.3, 2.0, 1.5, 17.1, 13.8, 14.1, 3.8, 2.2, 3.3, 3.0, 27.6, 1.8…
$ AREA <dbl> 0.1, 0.5, 0.5, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 2.0, 2.0, 2.…
$ YR.ISOL <dbl> 1968, 1920, 1900, 1966, 1918, 1965, 1955, 1920, 1965, 1900, 19…
$ DIST <dbl> 39, 234, 104, 66, 246, 234, 467, 284, 156, 311, 66, 93, 39, 40…
$ LDIST <dbl> 39, 234, 311, 66, 246, 285, 467, 1829, 156, 571, 332, 93, 39, …
$ GRAZE <dbl> 2, 5, 5, 3, 5, 3, 5, 5, 4, 5, 3, 5, 2, 1, 5, 5, 3, 3, 3, 2, 2,…
$ ALT <dbl> 160, 60, 140, 160, 140, 130, 90, 60, 130, 130, 210, 160, 210, …spc_tbl_ [56 × 7] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
$ ABUND : num [1:56] 5.3 2 1.5 17.1 13.8 14.1 3.8 2.2 3.3 3 ...
$ AREA : num [1:56] 0.1 0.5 0.5 1 1 1 1 1 1 1 ...
$ YR.ISOL: num [1:56] 1968 1920 1900 1966 1918 ...
$ DIST : num [1:56] 39 234 104 66 246 234 467 284 156 311 ...
$ LDIST : num [1:56] 39 234 311 66 246 ...
$ GRAZE : num [1:56] 2 5 5 3 5 3 5 5 4 5 ...
$ ALT : num [1:56] 160 60 140 160 140 130 90 60 130 130 ...
- attr(*, "spec")=
.. cols(
.. ABUND = col_double(),
.. AREA = col_double(),
.. YR.ISOL = col_double(),
.. DIST = col_double(),
.. LDIST = col_double(),
.. GRAZE = col_double(),
.. ALT = col_double()
.. )
- attr(*, "problems")=<externalptr> # A tibble: 6 × 7
ABUND AREA YR.ISOL DIST LDIST GRAZE ALT
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 5.3 0.1 1968 39 39 2 160
2 2 0.5 1920 234 234 5 60
3 1.5 0.5 1900 104 311 5 140
4 17.1 1 1966 66 66 3 160
5 13.8 1 1918 246 246 5 140
6 14.1 1 1965 234 285 3 130loyn (56 rows and 7 variables, 7 shown)
ID | Name | Type | Missings | Values | N
---+---------+---------+----------+--------------+-----------
1 | ABUND | numeric | 0 (0.0%) | [1.5, 39.6] | 56
---+---------+---------+----------+--------------+-----------
2 | AREA | numeric | 0 (0.0%) | [0.1, 1771] | 56
---+---------+---------+----------+--------------+-----------
3 | YR.ISOL | numeric | 0 (0.0%) | [1890, 1976] | 56
---+---------+---------+----------+--------------+-----------
4 | DIST | numeric | 0 (0.0%) | [26, 1427] | 56
---+---------+---------+----------+--------------+-----------
5 | LDIST | numeric | 0 (0.0%) | [26, 4426] | 56
---+---------+---------+----------+--------------+-----------
6 | GRAZE | numeric | 0 (0.0%) | 1 | 13 (23.2%)
| | | | 2 | 8 (14.3%)
| | | | 3 | 15 (26.8%)
| | | | 4 | 7 (12.5%)
| | | | 5 | 13 (23.2%)
---+---------+---------+----------+--------------+-----------
7 | ALT | numeric | 0 (0.0%) | [60, 260] | 56
-------------------------------------------------------------4 Exploratory data analysis
When we explored this analysis from a frequentist perspective, we decided on a log-normal like model. This was a model that was fit against a Gaussian distribution, yet with a log-link. We will replicate that model here in a Bayesian framework.
In the previous exploration of this model, we elected to treat Grazing intensity as a categorical variable - we will again code Grazing intensity as a categorical variable.
Model formula: \[ y_i \sim{} \mathcal{N}(\mu_i, \sigma^2)\\ log(\mu_i) = \boldsymbol{\beta} \bf{X_i}\\ \beta_0 \sim{} \mathcal{N}(3,0.5)\\ \beta_{1-9} \sim{} \mathcal{N}(0,2.5)\\ \sigma \sim{} \mathcal{Gamma}(2,1)\\ OR\\ \sigma \sim{} \mathcal{t}(3,0,2.5) \]
where \(\boldsymbol{\beta}\) is a vector of effects parameters and \(\bf{X}\) is a model matrix representing the additive effects of the scaled versions of distance (ln), distance to the nearest large patch (ln), patch area (ln), grazing intensity, year of isolation and altitude on the abundance of forest birds.
4.1 Scatterplot matrix
To re-acquaint ourselves with the data, I we will revisit the scatterplot matrix that we generated prior to the frequentist analysis.
scatterplotMatrix(~ABUND+DIST+LDIST+AREA+GRAZE+ALT+YR.ISOL, data = loyn,
diagonal = list(method = 'boxplot'))
We again notice that DIST, LDIST and AREA are skewed, so we will normalise them via a logarithmic transformation.
5 Fit the model
loyn.glm <- glm(ABUND~scale(log(DIST))+scale(log(LDIST))+scale(log(AREA))+
fGRAZE + scale(ALT) + scale(YR.ISOL),
data = loyn,
family = gaussian(link='log'))
loyn.glm |> summary()
Call:
glm(formula = ABUND ~ scale(log(DIST)) + scale(log(LDIST)) +
scale(log(AREA)) + fGRAZE + scale(ALT) + scale(YR.ISOL),
family = gaussian(link = "log"), data = loyn)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.092248 0.112856 27.400 < 2e-16 ***
scale(log(DIST)) -0.018067 0.057247 -0.316 0.75373
scale(log(LDIST)) 0.057086 0.059984 0.952 0.34623
scale(log(AREA)) 0.203976 0.059620 3.421 0.00132 **
fGRAZE2 0.039644 0.148978 0.266 0.79134
fGRAZE3 0.019654 0.137752 0.143 0.88717
fGRAZE4 -0.001197 0.156199 -0.008 0.99392
fGRAZE5 -1.121563 0.343631 -3.264 0.00208 **
scale(ALT) -0.003237 0.048607 -0.067 0.94719
scale(YR.ISOL) -0.018246 0.074404 -0.245 0.80737
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for gaussian family taken to be 42.40928)
Null deviance: 6337.9 on 55 degrees of freedom
Residual deviance: 1950.8 on 46 degrees of freedom
AIC: 379.76
Number of Fisher Scoring iterations: 6In rstanarm, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.
loyn.rstanarm = stan_glm(ABUND ~ scale(log(DIST))+
scale(log(LDIST))+
scale(log(AREA))+
fGRAZE+
scale(ALT)+
scale(YR.ISOL),
data=loyn,
family=gaussian(link='log'),
iter = 5000, warmup = 2500,
chains = 3, thin = 5, refresh = 0)Warning: There were 154 divergent transitions after warmup. See
https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
to find out why this is a problem and how to eliminate them.Warning: There were 2 chains where the estimated Bayesian Fraction of Missing Information was low. See
https://mc-stan.org/misc/warnings.html#bfmi-lowWarning: Examine the pairs() plot to diagnose sampling problemsWarning: The largest R-hat is 3.91, indicating chains have not mixed.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#r-hatWarning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#bulk-essWarning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#tail-essWarning: Markov chains did not converge! Do not analyze results!Conclusions:
- the warning messages suggest that this model has not performed well
- there are a large number of divergent transitions. Divergent transitions occur when the sampler is in a ‘sharp’ part of the posterior and the step length is too large causing the sampler to shoot off the posterior and reject the sample. This leads to inefficient and ineffective MCMC sampling. To alleviate this, we can either:
- review the model itself - it might be misspecified
- adjust the adaptive delta - a parameter that governs the degree of step-length learning that occurs during the warmup stage - the more it learns the less likely the sampler will be to overshoot (although it will take longer to learn)
- review the priors
- the largest R-hat value is large. This suggests that the chains have not converged well
- the effective sample sizes is too low, suggesting that there might be issues with the priors.
- there are a large number of divergent transitions. Divergent transitions occur when the sampler is in a ‘sharp’ part of the posterior and the step length is too large causing the sampler to shoot off the posterior and reject the sample. This leads to inefficient and ineffective MCMC sampling. To alleviate this, we can either:
loyn.rstanarm = stan_glm(ABUND ~ scale(log(DIST))+
scale(log(LDIST))+
scale(log(AREA))+
fGRAZE+
scale(ALT)+
scale(YR.ISOL),
data=loyn,
family=gaussian(link='log'),
iter = 5000, warmup = 2500,
chains = 3, thin = 5, refresh = 0,
adapt_delta = 0.99)Warning: There were 1039 divergent transitions after warmup. See
https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
to find out why this is a problem and how to eliminate them.Warning: There were 3 chains where the estimated Bayesian Fraction of Missing Information was low. See
https://mc-stan.org/misc/warnings.html#bfmi-lowWarning: Examine the pairs() plot to diagnose sampling problemsWarning: The largest R-hat is 5.98, indicating chains have not mixed.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#r-hatWarning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#bulk-essWarning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#tail-essWarning: Markov chains did not converge! Do not analyze results!There are still a substantial number of divergent transitions. It is possible that the priors are too vague.
Priors for model 'loyn.rstanarm'
------
Intercept (after predictors centered)
Specified prior:
~ normal(location = 0, scale = 2.5)
Adjusted prior:
~ normal(location = 0, scale = 27)
Coefficients
Specified prior:
~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
Adjusted prior:
~ normal(location = [0,0,0,...], scale = [26.84,26.84,26.84,...])
Auxiliary (sigma)
Specified prior:
~ exponential(rate = 1)
Adjusted prior:
~ exponential(rate = 0.093)
------
See help('prior_summary.stanreg') for more detailsConclusions:
- the default priors appear to be overly wide. We will instead define our own priors. ### Assessing priors
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Conclusions:
- we see that the range of predictions are extremely wide and the slope could range from strongly negative to strongly positive.
- \(\beta_0\): normal centred at 3 with a standard deviation of 1 (on the log scale, these are the equivalent of 20.1 and 1 respectively)
- mean of 3: since
mean(log(loyn$ABUND)) - sd of 1: since
sd(log(loyn$ABUND))
- mean of 3: since
- \(\beta_1\): normal centred at 0 with a standard deviation of 2.5 (again, consider what this would be on a response scale)
- sd of 1: since
sd(log(loyn$ABUND))/sd(scale(log(loyn$AREA)))(and since each predictor is scaled, it will be the same for each predictor)
- sd of 1: since
- \(\sigma\): half cauchy (flat Gaussian) prior with a scale of 2.
I will also overlay the raw data for comparison.
loyn.rstanarm2 <- stan_glm(ABUND ~ scale(log(DIST))+
scale(log(LDIST))+
scale(log(AREA))+
fGRAZE+
scale(ALT)+
scale(YR.ISOL), data=loyn,
family=gaussian(link='log'),
prior_intercept = normal(3, 1, autoscale=FALSE),
prior = normal(0, 1, autoscale=FALSE),
prior_aux = cauchy(0, 2),
prior_PD=TRUE,
iter = 5000, thin=5,
chains = 3, warmup=2500,
refresh=0) Conclusions:
- there are no longer any warning messages.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Now lets refit, conditioning on the data.
Drawing from prior...
Conclusions:
- in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.
#ggemmeans(loyn.rstanarm3, ~AREA) |> plot(add.data=TRUE) + scale_y_log10()
ggpredict(loyn.rstanarm3, terms = "AREA[0:1000]") |>
plot(jitter = FALSE, add.data=TRUE, log.y = TRUE) + scale_x_log10()Warning: Transformation introduced infinite values in continuous x-axis
Transformation introduced infinite values in continuous x-axisWarning: Removed 1 row containing missing values (`geom_line()`).
## ggpredict(loyn.rstanarm3, terms = "AREA[0:1000]") |>
## plot(jitter = FALSE, residuals=TRUE, log.y = TRUE) + scale_x_log10()
ggpredict(loyn.rstanarm3) |>
plot(add.data = TRUE) |>
wrap_plots() &
scale_y_log10()Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Note that for the second of the above are partial plots (effect of one predictor at the average of the continuous predictors and the first level of the categorical predictor), the raw data are not similarly standardised and thus may appear not to match the trends…
In brms, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.
Unlike rstanarm, brms models must be compiled before they start sampling. For most models, the compilation of the stan code takes around 45 seconds.
loyn.form <- bf(ABUND ~ scale(log(DIST))+
scale(log(LDIST))+
scale(log(AREA))+
fGRAZE+
scale(ALT)+
scale(YR.ISOL),
family = gaussian(link = 'log'))
loyn.brm <- brm(loyn.form,
data = loyn,
iter = 5000,
warmup = 2500,
chains = 3, cores = 3,
thin = 5,
refresh = 0,
backend = "cmdstan")Start samplingRunning MCMC with 3 parallel chains...
Chain 1 finished in 0.9 seconds.
Chain 3 finished in 1.8 seconds.
Chain 2 finished in 2.0 seconds.
All 3 chains finished successfully.
Mean chain execution time: 1.6 seconds.
Total execution time: 2.2 seconds. prior class coef group resp dpar nlpar lb ub source
(flat) b default
(flat) b fGRAZE2 (vectorized)
(flat) b fGRAZE3 (vectorized)
(flat) b fGRAZE4 (vectorized)
(flat) b fGRAZE5 (vectorized)
(flat) b scaleALT (vectorized)
(flat) b scalelogAREA (vectorized)
(flat) b scalelogDIST (vectorized)
(flat) b scalelogLDIST (vectorized)
(flat) b scaleYR.ISOL (vectorized)
student_t(3, 3, 2.5) Intercept default
student_t(3, 0, 11.8) sigma 0 defaultpriors <- prior(normal(0, 2.5), class = 'b')
loyn.form <- bf(ABUND ~ scale(log(DIST))+
scale(log(LDIST))+
scale(log(AREA))+
fGRAZE+
scale(ALT)+
scale(YR.ISOL),
family = gaussian(link = 'log'))
loyn.brm1 <- brm(loyn.form,
data = loyn,
prior = priors,
sample_prior = 'only',
iter = 5000,
warmup = 2500,
chains = 3,
thin = 5,
refresh = 0,
backend = "cmdstan")Start samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 0.1 seconds.
Chain 2 finished in 0.1 seconds.
Chain 3 finished in 0.1 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.1 seconds.
Total execution time: 0.5 seconds.## Individual plots - the following seems to be broken??
##loyn.brm1 |> ggemmeans(~AREA) |> plot(add.data = TRUE) + scale_y_log10()
loyn.brm1 |>
ggemmeans(~AREA) |>
plot(add.data = TRUE) + scale_y_log10()Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
## All effects
loyn.brm1 |>
conditional_effects() |>
plot(points = TRUE, ask = FALSE, plot = FALSE) |>
wrap_plots() &
scale_y_log10()
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.Warning: Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Conclusions:
- we see that the range of predictions is fairly wide and the slope could range from strongly negative to strongly positive.
- \(\beta_0\): normal centred at 3 with a standard deviation of 1 (on the log scale, these are the equivalent of 20.1 and 1 respectively)
- mean of 3: since
median(log(loyn$ABUND)) - sd of 0.5: since
mad(log(loyn$ABUND))
- mean of 3: since
- \(\beta_1\): normal centred at 0 with a standard deviation of 2.5 (again, consider what this would be on a response scale)
- sd of 1: since
mad(log(loyn$ABUND))/mad(scale(log(loyn$AREA)))(and since each predictor is scaled, it will be the same for each predictor)
- sd of 1: since
- \(\sigma\): half cauchy (flat Gaussian) prior with a scale of 1.
mod.mat <- model.matrix(as.formula(loyn.form), data = loyn)
mad(log(loyn$ABUND))/
apply(mod.mat, 2, mad) (Intercept) scale(log(DIST)) scale(log(LDIST)) scale(log(AREA))
Inf 0.6280229 0.5626313 0.5009688
fGRAZE2 fGRAZE3 fGRAZE4 fGRAZE5
Inf Inf Inf Inf
scale(ALT) scale(YR.ISOL)
0.5127373 1.3907494 priors <- prior(normal(3, 0.5), class = 'Intercept') +
prior(normal(0, 1.5), class = 'b') +
## prior(gamma(1, 1), class = 'sigma')
prior(student_t(3, 0, 2.5), class = 'sigma')
loyn.form <- bf(ABUND ~ scale(log(DIST))+
scale(log(LDIST))+
scale(log(AREA))+
fGRAZE+
scale(ALT)+
scale(YR.ISOL),
family = gaussian(link = 'log'))
## family = lognormal())
loyn.brm2 <- brm(loyn.form,
data = loyn,
prior = priors,
sample_prior = 'only',
iter = 5000,
warmup = 2500,
chains = 3, cores = 3,
thin = 5,
refresh = 0,
backend = "cmdstan")Start samplingRunning MCMC with 3 parallel chains...
Chain 1 finished in 0.0 seconds.
Chain 2 finished in 0.1 seconds.
Chain 3 finished in 0.0 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.0 seconds.
Total execution time: 0.2 seconds.Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
loyn.brm2 |>
conditional_effects() |>
plot(points = TRUE, ask = FALSE, plot = FALSE) |>
## lapply(function(x) x + scale_y_log10()) |>
wrap_plots() &
scale_y_log10()
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.Warning: Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
The desired updates require recompiling the modelStart samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 0.3 seconds.
Chain 2 finished in 0.3 seconds.
Chain 3 finished in 0.3 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.3 seconds.
Total execution time: 1.3 seconds. [1] "b_Intercept" "b_scalelogDIST" "b_scalelogLDIST" "b_scalelogAREA"
[5] "b_fGRAZE2" "b_fGRAZE3" "b_fGRAZE4" "b_fGRAZE5"
[9] "b_scaleALT" "b_scaleYR.ISOL" "sigma" "prior_Intercept"
[13] "prior_b" "prior_sigma" "lprior" "lp__"
[17] "accept_stat__" "treedepth__" "stepsize__" "divergent__"
[21] "n_leapfrog__" "energy__" ## loyn.brm3 |> hypothesis('Intercept = 0', class = 'b') |> plot
## loyn.brm3 |> hypothesis('Intercept = 0', class = 'prior') |> plot
loyn.brm3 |> hypothesis('scalelogDIST = 0') |> plot()
$N
[1] 56
$Y
[1] 5.3 2.0 1.5 17.1 13.8 14.1 3.8 2.2 3.3 3.0 27.6 1.8 21.2 14.6 8.0
[16] 3.5 29.0 2.9 24.3 19.4 24.4 5.0 15.8 25.3 19.5 20.9 16.3 18.8 19.9 13.0
[31] 6.8 21.7 27.8 26.8 16.6 30.4 11.5 26.0 25.7 12.7 23.5 24.9 29.0 28.3 28.3
[46] 32.0 37.7 39.6 29.6 31.0 34.4 27.3 30.5 33.0 29.5 30.9
$K
[1] 10
$X
Intercept scalelogDIST scalelogLDIST scalelogAREA fGRAZE2 fGRAZE3 fGRAZE4
1 1 -1.51019511 -1.65892116 -2.37802367 1 0 0
2 1 0.37029448 -0.30532977 -1.51765965 0 0 0
3 1 -0.48079400 -0.09042449 -1.51765965 0 0 0
4 1 -0.95804924 -1.26148217 -1.14712103 0 1 0
5 1 0.42278148 -0.26754922 -1.14712103 0 0 0
6 1 0.37029448 -0.15637842 -1.14712103 0 1 0
7 1 1.09552219 0.21669491 -1.14712103 0 0 0
8 1 0.57353754 1.24803687 -1.14712103 0 0 0
9 1 -0.05524976 -0.61163991 -1.14712103 0 0 1
10 1 0.66885367 0.36858640 -1.14712103 0 0 0
11 1 -0.95804924 -0.04106159 -0.77658241 0 1 0
12 1 -0.59812145 -1.00240327 -0.77658241 0 0 0
13 1 -1.51019511 -1.65892116 -0.77658241 1 0 0
14 1 0.93822292 0.10346964 -0.77658241 0 0 0
15 1 0.47682817 -0.22864597 -0.77658241 0 0 0
16 1 -0.24660010 0.43442972 -0.77658241 0 0 0
17 1 -1.93573935 -1.96523129 -0.55983122 0 1 0
18 1 -1.93573935 -1.96523129 -0.55983122 0 1 0
19 1 -1.48362354 -1.63979473 -0.40604380 0 1 0
20 1 0.47682817 -0.22864597 -0.40604380 1 0 0
21 1 0.37029448 0.29645165 -0.40604380 1 0 0
22 1 -1.93573935 1.38927509 -0.40604380 0 0 0
23 1 -1.51019511 -1.65892116 -0.28675700 0 1 0
24 1 0.85682391 0.04487798 -0.28675700 0 0 0
25 1 -0.59812145 -0.33160908 -0.18929260 0 1 0
26 1 -0.03525827 0.79868571 -0.18929260 0 1 0
27 1 0.57722655 0.69705984 -0.10688762 0 1 0
28 1 -0.22265561 -0.73213997 -0.10688762 0 0 1
29 1 0.50481706 -0.20849935 -0.03550518 0 0 1
30 1 0.79284436 1.26397616 0.02745860 0 0 0
31 1 0.75311975 0.29645165 0.08378161 0 1 0
32 1 -1.89613008 -1.93672022 0.13473198 0 0 1
33 1 -0.03525827 -0.59724988 0.18124602 0 0 1
34 1 -0.22265561 -0.73213997 0.18124602 0 0 1
35 1 -0.12477989 -0.66168825 0.22403479 1 0 0
36 1 -0.12477989 0.09591504 0.30053281 0 1 0
37 1 0.90372228 1.51230754 0.36744180 0 0 0
38 1 -1.48362354 1.66778539 0.39799721 1 0 0
39 1 0.50481706 0.99128245 0.42690016 0 0 1
40 1 0.66885367 1.53439756 0.50527059 0 0 0
41 1 1.35327186 0.40222517 0.59457340 0 0 0
42 1 1.25762760 0.59446805 0.65294853 0 1 0
43 1 0.24667868 -0.39430941 0.70557205 0 0 0
44 1 -0.95804924 -0.01204503 0.73798041 0 0 0
45 1 0.57722655 -0.51197475 0.82485885 1 0 0
46 1 -0.59812145 -1.00240327 0.87580921 0 1 0
47 1 0.47682817 0.98837574 0.92232325 0 1 0
48 1 2.26783778 1.12640241 0.93334579 0 0 0
49 1 0.92772762 1.07832552 0.94414564 0 0 0
50 1 1.09552219 0.21669491 1.01418997 0 0 0
51 1 -1.51019511 0.29645165 1.29286187 1 0 0
52 1 0.93822292 1.91565164 1.35582564 0 0 0
53 1 1.09552219 1.39201101 1.47113789 0 0 0
54 1 -0.12477989 -0.07123733 1.50961306 0 0 0
55 1 0.75311975 1.00336997 2.53095496 0 0 0
56 1 0.73743154 -0.04106159 2.85111978 0 0 0
fGRAZE5 scaleALT scaleYR.ISOL
1 0 0.31588146 0.7134084
2 1 -1.98143824 -1.1629534
3 1 -0.14358248 -1.9447708
4 0 0.31588146 0.6352266
5 1 -0.14358248 -1.2411351
6 0 -0.37331445 0.5961358
7 1 -1.29224233 0.2052271
8 1 -1.98143824 -1.1629534
9 0 -0.37331445 0.5961358
10 1 -0.37331445 -1.9447708
11 0 1.46454131 -0.9284082
12 1 0.31588146 -2.3356795
13 0 1.46454131 0.9088627
14 0 1.46454131 0.8697719
15 1 -0.60304642 -1.9447708
16 1 -0.02871650 -1.9447708
17 0 -0.83277839 0.4788632
18 0 -0.14358248 0.5961358
19 0 1.00507737 0.4006814
20 0 -1.29224233 0.1270453
21 0 1.69427328 0.9088627
22 1 -0.60304642 -1.0456808
23 0 -0.37331445 0.5961358
24 0 -1.06251036 0.6743175
25 0 0.54561343 -2.3356795
26 0 0.08614949 0.4006814
27 0 -0.37331445 0.5961358
28 0 1.46454131 0.4006814
29 0 -0.60304642 0.9088627
30 1 -1.29224233 -1.5538621
31 0 -0.83277839 0.4788632
32 0 0.66047941 0.4006814
33 0 -0.83277839 0.5179540
34 0 1.23480934 0.4006814
35 0 1.00507737 0.7134084
36 0 -0.60304642 0.6352266
37 1 -1.06251036 -1.1629534
38 0 1.00507737 0.6352266
39 0 0.08614949 0.9088627
40 1 -1.29224233 -1.2411351
41 0 -0.14358248 0.5179540
42 0 -0.37331445 0.5961358
43 0 1.00507737 0.9479536
44 0 -0.83277839 0.5961358
45 0 -0.60304642 0.4788632
46 0 1.00507737 0.4006814
47 0 -0.60304642 -0.8502264
48 0 0.77534540 0.8697719
49 0 -0.14358248 0.6743175
50 0 0.43074744 0.5179540
51 0 0.66047941 1.0261353
52 0 -1.75170627 0.5570449
53 0 0.31588146 0.5570449
54 0 1.00507737 -0.3811360
55 0 1.00507737 0.7915901
56 0 2.61320116 -0.6547721
attr(,"assign")
[1] 0 1 2 3 4 4 4 4 5 6
attr(,"contrasts")
attr(,"contrasts")$fGRAZE
2 3 4 5
1 0 0 0 0
2 1 0 0 0
3 0 1 0 0
4 0 0 1 0
5 0 0 0 1
$prior_only
[1] 0
attr(,"class")
[1] "standata" "list" // generated with brms 2.19.0
functions {
}
data {
int<lower=1> N; // total number of observations
vector[N] Y; // response variable
int<lower=1> K; // number of population-level effects
matrix[N, K] X; // population-level design matrix
int prior_only; // should the likelihood be ignored?
}
transformed data {
int Kc = K - 1;
matrix[N, Kc] Xc; // centered version of X without an intercept
vector[Kc] means_X; // column means of X before centering
for (i in 2 : K) {
means_X[i - 1] = mean(X[ : , i]);
Xc[ : , i - 1] = X[ : , i] - means_X[i - 1];
}
}
parameters {
vector[Kc] b; // population-level effects
real Intercept; // temporary intercept for centered predictors
real<lower=0> sigma; // dispersion parameter
}
transformed parameters {
real lprior = 0; // prior contributions to the log posterior
lprior += normal_lpdf(b | 0, 1.5);
lprior += normal_lpdf(Intercept | 3, 0.5);
lprior += student_t_lpdf(sigma | 3, 0, 2.5)
- 1 * student_t_lccdf(0 | 3, 0, 2.5);
}
model {
// likelihood including constants
if (!prior_only) {
// initialize linear predictor term
vector[N] mu = rep_vector(0.0, N);
mu += Intercept + Xc * b;
mu = exp(mu);
target += normal_lpdf(Y | mu, sigma);
}
// priors including constants
target += lprior;
}
generated quantities {
// actual population-level intercept
real b_Intercept = Intercept - dot_product(means_X, b);
// additionally sample draws from priors
real prior_b = normal_rng(0, 1.5);
real prior_Intercept = normal_rng(3, 0.5);
real prior_sigma = student_t_rng(3, 0, 2.5);
// use rejection sampling for truncated priors
while (prior_sigma < 0) {
prior_sigma = student_t_rng(3, 0, 2.5);
}
}6 MCMC sampling diagnostics
The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.
See list of available diagnostics by name
bayesplot MCMC module:
mcmc_acf
mcmc_acf_bar
mcmc_areas
mcmc_areas_ridges
mcmc_combo
mcmc_dens
mcmc_dens_chains
mcmc_dens_overlay
mcmc_hex
mcmc_hist
mcmc_hist_by_chain
mcmc_intervals
mcmc_neff
mcmc_neff_hist
mcmc_nuts_acceptance
mcmc_nuts_divergence
mcmc_nuts_energy
mcmc_nuts_stepsize
mcmc_nuts_treedepth
mcmc_pairs
mcmc_parcoord
mcmc_rank_ecdf
mcmc_rank_hist
mcmc_rank_overlay
mcmc_recover_hist
mcmc_recover_intervals
mcmc_recover_scatter
mcmc_rhat
mcmc_rhat_hist
mcmc_scatter
mcmc_trace
mcmc_trace_highlight
mcmc_violinOf these, we will focus on:
- mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
Warning: The `facets` argument of `facet_grid()` is deprecated as of ggplot2 2.2.0.
ℹ Please use the `rows` argument instead.
ℹ The deprecated feature was likely used in the bayesplot package.
Please report the issue at <https://github.com/stan-dev/bayesplot/issues/>.
There is no evidence of auto-correlation in the MCMC samples
- Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
All Rhat values are below 1.05, suggesting the chains have converged.
neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Ratios all very high.
The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- stan_acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
There is no evidence of auto-correlation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Ratios all very high.
The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Warning: Returning more (or less) than 1 row per `summarise()` group was deprecated in
dplyr 1.1.0.
ℹ Please use `reframe()` instead.
ℹ When switching from `summarise()` to `reframe()`, remember that `reframe()`
always returns an ungrouped data frame and adjust accordingly.
ℹ The deprecated feature was likely used in the ggmcmc package.
Please report the issue at <https://github.com/xfim/ggmcmc/issues/>.
Ratios all very high.
The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.
See list of available diagnostics by name
bayesplot MCMC module:
mcmc_acf
mcmc_acf_bar
mcmc_areas
mcmc_areas_ridges
mcmc_combo
mcmc_dens
mcmc_dens_chains
mcmc_dens_overlay
mcmc_hex
mcmc_hist
mcmc_hist_by_chain
mcmc_intervals
mcmc_neff
mcmc_neff_hist
mcmc_nuts_acceptance
mcmc_nuts_divergence
mcmc_nuts_energy
mcmc_nuts_stepsize
mcmc_nuts_treedepth
mcmc_pairs
mcmc_parcoord
mcmc_rank_ecdf
mcmc_rank_hist
mcmc_rank_overlay
mcmc_recover_hist
mcmc_recover_intervals
mcmc_recover_scatter
mcmc_rhat
mcmc_rhat_hist
mcmc_scatter
mcmc_trace
mcmc_trace_highlight
mcmc_violinOf these, we will focus on:
- mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
Warning: The `facets` argument of `facet_grid()` is deprecated as of ggplot2 2.2.0.
ℹ Please use the `rows` argument instead.
ℹ The deprecated feature was likely used in the bayesplot package.
Please report the issue at <https://github.com/stan-dev/bayesplot/issues/>.
There is no evidence of autocorrelation in the MCMC samples
- Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
All Rhat values are below 1.05, suggesting the chains have converged.
neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Ratios all very high.
The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Ratios all very high.
The ggmean package also has a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
The chains appear well mixed and very similar
- gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
Warning: Returning more (or less) than 1 row per `summarise()` group was deprecated in
dplyr 1.1.0.
ℹ Please use `reframe()` instead.
ℹ When switching from `summarise()` to `reframe()`, remember that `reframe()`
always returns an ungrouped data frame and adjust accordingly.
ℹ The deprecated feature was likely used in the ggmcmc package.
Please report the issue at <https://github.com/xfim/ggmcmc/issues/>.
Ratios all very high.
7 Model validation
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
See list of available diagnostics by name
bayesplot PPC module:
ppc_bars
ppc_bars_grouped
ppc_boxplot
ppc_dens
ppc_dens_overlay
ppc_dens_overlay_grouped
ppc_ecdf_overlay
ppc_ecdf_overlay_grouped
ppc_error_binned
ppc_error_hist
ppc_error_hist_grouped
ppc_error_scatter
ppc_error_scatter_avg
ppc_error_scatter_avg_grouped
ppc_error_scatter_avg_vs_x
ppc_freqpoly
ppc_freqpoly_grouped
ppc_hist
ppc_intervals
ppc_intervals_grouped
ppc_km_overlay
ppc_km_overlay_grouped
ppc_loo_intervals
ppc_loo_pit
ppc_loo_pit_overlay
ppc_loo_pit_qq
ppc_loo_ribbon
ppc_pit_ecdf
ppc_pit_ecdf_grouped
ppc_ribbon
ppc_ribbon_grouped
ppc_rootogram
ppc_scatter
ppc_scatter_avg
ppc_scatter_avg_grouped
ppc_stat
ppc_stat_2d
ppc_stat_freqpoly
ppc_stat_freqpoly_grouped
ppc_stat_grouped
ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
The model draws appear deviate from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.
- error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.
- intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
The modelled predictions seem to underestimate the uncertainty with increasing mussel clump area.
- ribbon: this is just an alternative way of expressing the above plot.
The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.
DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- posterior_predict(loyn.rstanarm3, ndraws=250, summary=FALSE)
loyn.resids <- createDHARMa(simulatedResponse = t(preds),
observedResponse = loyn$ABUND,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = FALSE)
plot(loyn.resids)
Conclusions:
- the simulated residuals suggest a general lack of fit due to over dispersion and outliers
- perhaps we should explore a negative binomial model
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
See list of available diagnostics by name
bayesplot PPC module:
ppc_bars
ppc_bars_grouped
ppc_boxplot
ppc_dens
ppc_dens_overlay
ppc_dens_overlay_grouped
ppc_ecdf_overlay
ppc_ecdf_overlay_grouped
ppc_error_binned
ppc_error_hist
ppc_error_hist_grouped
ppc_error_scatter
ppc_error_scatter_avg
ppc_error_scatter_avg_grouped
ppc_error_scatter_avg_vs_x
ppc_freqpoly
ppc_freqpoly_grouped
ppc_hist
ppc_intervals
ppc_intervals_grouped
ppc_km_overlay
ppc_km_overlay_grouped
ppc_loo_intervals
ppc_loo_pit
ppc_loo_pit_overlay
ppc_loo_pit_qq
ppc_loo_ribbon
ppc_pit_ecdf
ppc_pit_ecdf_grouped
ppc_ribbon
ppc_ribbon_grouped
ppc_rootogram
ppc_scatter
ppc_scatter_avg
ppc_scatter_avg_grouped
ppc_stat
ppc_stat_2d
ppc_stat_freqpoly
ppc_stat_freqpoly_grouped
ppc_stat_grouped
ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
The model draws appear deviate from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
Using all posterior draws for ppc type 'error_scatter_avg' by default.
The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.
- error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.
Using all posterior draws for ppc type 'error_scatter_avg_vs_x' by default.
- intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
Using all posterior draws for ppc type 'intervals' by default.
The modelled predictions seem to underestimate the uncertainty with increasing mussel clump area.
- ribbon: this is just an alternative way of expressing the above plot.
Using all posterior draws for ppc type 'ribbon' by default.
The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.
DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- loyn.brm3 |> posterior_predict(ndraws = 250, summary = FALSE)
loyn.resids <- createDHARMa(simulatedResponse = t(preds),
observedResponse = loyn$ABUND,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = FALSE)
loyn.resids |> plot()
loyn.resids <- make_brms_dharma_res(loyn.brm3, integerResponse = FALSE)
wrap_elements(~testUniformity(loyn.resids)) +
wrap_elements(~plotResiduals(loyn.resids, form = factor(rep(1, nrow(loyn))))) +
wrap_elements(~plotResiduals(loyn.resids, quantreg = TRUE)) +
wrap_elements(~testDispersion(loyn.resids))
Conclusions:
- the simulated residuals suggest a general lack of fit due to over-dispersion and outliers
- perhaps we should explore a negative binomial model
8 Partial effects plots
loyn.rstanarm3 |> fitted_draws(newdata=loyn) |>
median_hdci() |>
ggplot(aes(x=AREA, y=.value)) +
geom_ribbon(aes(ymin=.lower, ymax=.upper), fill='blue', alpha=0.3) +
geom_line() +
geom_point(data=loyn, aes(y=ABUND, x=AREA)) +
scale_y_log10()Warning: `fitted_draws` and `add_fitted_draws` are deprecated as their names were confusing.
Use [add_]epred_draws() to get the expectation of the posterior predictive.
Use [add_]linpred_draws() to get the distribution of the linear predictor.
For example, you used [add_]fitted_draws(..., scale = "response"), which
means you most likely want [add_]epred_draws(...).
loyn.brm3 |>
conditional_effects() |>
plot(ask = FALSE, points = TRUE, plot = FALSE) |>
wrap_plots()
loyn.brm3 |>
conditional_effects() |>
plot(ask = FALSE, points = TRUE, plot = FALSE) |>
wrap_plots() &
scale_y_log10()
g <- loyn.brm3 |>
conditional_effects() |>
plot(ask = FALSE, points = TRUE, plot = FALSE)
library(patchwork)
length(g)[1] 6(g[[1]] + scale_x_log10()) +
(g[[2]] + scale_x_log10()) +
(g[[3]] + scale_x_log10()) +
g[[4]] +
g[[5]] +
g[[6]]
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.Warning: Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
Transformation introduced infinite values in continuous y-axis
This will be slow …
loyn.brm3 |>
epred_draws(newdata = loyn) |>
median_hdci(.epred) |>
ggplot(aes(x = AREA, y = .epred, colour = fGRAZE, fill = fGRAZE)) +
geom_ribbon(aes(ymin = .lower, ymax = .upper), colour = NA, alpha = 0.3) +
geom_line() +
geom_point(data = loyn, aes(y = ABUND, x = AREA)) +
scale_y_log10() +
scale_x_log10()
9 Model investigation
The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).
Model Info:
function: stan_glm
family: gaussian [log]
formula: ABUND ~ scale(log(DIST)) + scale(log(LDIST)) + scale(log(AREA)) +
fGRAZE + scale(ALT) + scale(YR.ISOL)
algorithm: sampling
sample: 1500 (posterior sample size)
priors: see help('prior_summary')
observations: 56
predictors: 10
Estimates:
mean sd 10% 50% 90%
(Intercept) 3.0 0.1 2.9 3.0 3.2
scale(log(DIST)) 0.0 0.1 -0.1 0.0 0.1
scale(log(LDIST)) 0.0 0.1 0.0 0.1 0.1
scale(log(AREA)) 0.2 0.1 0.1 0.2 0.3
fGRAZE2 0.0 0.2 -0.2 0.0 0.2
fGRAZE3 0.0 0.1 -0.1 0.0 0.2
fGRAZE4 0.0 0.2 -0.2 0.0 0.2
fGRAZE5 -1.1 0.4 -1.5 -1.1 -0.6
scale(ALT) 0.0 0.1 -0.1 0.0 0.1
scale(YR.ISOL) 0.0 0.1 -0.1 0.0 0.1
sigma 6.6 0.7 5.8 6.5 7.5
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 19.3 1.2 17.7 19.4 20.9
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 1305
scale(log(DIST)) 0.0 1.0 1288
scale(log(LDIST)) 0.0 1.0 1497
scale(log(AREA)) 0.0 1.0 1502
fGRAZE2 0.0 1.0 1507
fGRAZE3 0.0 1.0 1255
fGRAZE4 0.0 1.0 1434
fGRAZE5 0.0 1.0 1477
scale(ALT) 0.0 1.0 1374
scale(YR.ISOL) 0.0 1.0 1497
sigma 0.0 1.0 1377
mean_PPD 0.0 1.0 1531
log-posterior 0.1 1.0 1412
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).Conclusions:
- in the Model Info, we are informed that the total MCMC posterior sample size is 1500 and that there were 56 raw observations.
- the estimated intercept (expected bird abundance when grazing intensity is equal to 1 and all of the continuous predictors are at their respective averages) is 3.05. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 21.02.
- the estimated slope associated with
DIST(rate at which the abundance of birds changes per 1 unit change in log-transformed Distance to nearest patch holding other continuous predictors constant and grazing intensity at level 1), is -0.01 (mean) or -0.08 (median) with a standard deviation of 0. The 90% credibility intervals indicate that we are 90% confident that the slope is between -0.01 and -0.01 - e.g. there is no evidence of a significant trend.
- associated with each of the continuous predictors is a partial slope. Each partial slope is the rate of change between bird abundance and the associated predictor (on the log scale due to the link and based on 1 unit change in the predictor on the scale of the predictor). For example, for every one unit change in centred log patch Area, bird abundance is expected to increase by (log) 0.22.
- if we back transform this slope (by exponentiation), we get a partial slope for centred log Area of 1.24. This is interpreted as - for every 1 unit increase in (scaled log) Area, the bird abundance is expected to increase 1.24 fold. That is, there is a 24 percent increase per 1 unit increase in centred log Area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
loyn.rstanarm3$stanfit |>
summarise_draws(median,
HDInterval::hdi,
rhat, length, ess_bulk, ess_tail)# A tibble: 13 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <num> <num> <num> <num> <num> <num> <num>
1 (Intercept) 3.05e+0 2.84e+0 3.29 1.00 1500 1323. 1221.
2 scale(log(DIST)) -1.25e-2 -1.22e-1 0.108 1.00 1500 1288. 1082.
3 scale(log(LDIST)) 5.12e-2 -8.02e-2 0.161 1.00 1500 1482. 1414.
4 scale(log(AREA)) 2.13e-1 9.47e-2 0.333 0.999 1500 1518. 1334.
5 fGRAZE2 3.61e-2 -2.80e-1 0.337 1.00 1500 1546. 1498.
6 fGRAZE3 3.78e-2 -2.07e-1 0.330 1.00 1500 1246. 1389.
7 fGRAZE4 1.10e-2 -3.14e-1 0.322 1.00 1500 1417. 1216.
8 fGRAZE5 -1.06e+0 -1.79e+0 -0.438 1.00 1500 1494. 1419.
9 scale(ALT) -2.00e-3 -1.03e-1 0.104 1.00 1500 1384. 1544.
10 scale(YR.ISOL) 1.91e-4 -1.55e-1 0.157 1.00 1500 1485. 1350.
11 sigma 6.49e+0 5.37e+0 8.02 0.999 1500 1386. 1428.
12 mean_PPD 1.94e+1 1.67e+1 21.5 1.00 1500 1529. 1236.
13 log-posterior -1.96e+2 -2.02e+2 -192. 1.00 1500 1413. 1383.We can also alter the CI level.
loyn.rstanarm3$stanfit |>
summarise_draws(median,
~HDInterval::hdi(.x, credMass = 0.9),
rhat, length, ess_bulk, ess_tail)# A tibble: 13 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <num> <num> <num> <num> <num> <num> <num>
1 (Intercept) 3.05e+0 2.84e+0 3.23e+0 1.00 1500 1323. 1221.
2 scale(log(DIST)) -1.25e-2 -1.07e-1 8.53e-2 1.00 1500 1288. 1082.
3 scale(log(LDIST)) 5.12e-2 -4.48e-2 1.55e-1 1.00 1500 1482. 1414.
4 scale(log(AREA)) 2.13e-1 1.16e-1 3.16e-1 0.999 1500 1518. 1334.
5 fGRAZE2 3.61e-2 -2.16e-1 3.04e-1 1.00 1500 1546. 1498.
6 fGRAZE3 3.78e-2 -1.73e-1 2.74e-1 1.00 1500 1246. 1389.
7 fGRAZE4 1.10e-2 -2.72e-1 2.61e-1 1.00 1500 1417. 1216.
8 fGRAZE5 -1.06e+0 -1.60e+0 -4.94e-1 1.00 1500 1494. 1419.
9 scale(ALT) -2.00e-3 -8.37e-2 8.54e-2 1.00 1500 1384. 1544.
10 scale(YR.ISOL) 1.91e-4 -1.31e-1 1.26e-1 1.00 1500 1485. 1350.
11 sigma 6.49e+0 5.49e+0 7.65e+0 0.999 1500 1386. 1428.
12 mean_PPD 1.94e+1 1.73e+1 2.13e+1 1.00 1500 1529. 1236.
13 log-posterior -1.96e+2 -2.01e+2 -1.92e+2 1.00 1500 1413. 1383.And on a ratio scale
loyn.rstanarm3$stanfit |>
summarise_draws(
~ median(exp(.x)),
~HDInterval::hdi(exp(.x)),
rhat, length, ess_bulk, ess_tail)# A tibble: 13 × 8
variable `~median(exp(.x))` lower upper rhat length ess_bulk ess_tail
<chr> <num> <num> <num> <num> <num> <num> <num>
1 (Interce… 2.11e+ 1 1.66e+ 1 2.62e+ 1 1.00 1500 1323. 1221.
2 scale(lo… 9.88e- 1 8.80e- 1 1.11e+ 0 1.00 1500 1288. 1082.
3 scale(lo… 1.05e+ 0 9.23e- 1 1.17e+ 0 1.00 1500 1482. 1414.
4 scale(lo… 1.24e+ 0 1.10e+ 0 1.39e+ 0 0.999 1500 1518. 1334.
5 fGRAZE2 1.04e+ 0 7.56e- 1 1.40e+ 0 1.00 1500 1546. 1498.
6 fGRAZE3 1.04e+ 0 7.80e- 1 1.34e+ 0 1.00 1500 1246. 1389.
7 fGRAZE4 1.01e+ 0 7.05e- 1 1.35e+ 0 1.00 1500 1417. 1216.
8 fGRAZE5 3.47e- 1 1.43e- 1 6.05e- 1 1.00 1500 1494. 1419.
9 scale(AL… 9.98e- 1 8.95e- 1 1.10e+ 0 1.00 1500 1384. 1544.
10 scale(YR… 1.00e+ 0 8.50e- 1 1.16e+ 0 1.00 1500 1485. 1350.
11 sigma 6.58e+ 2 1.34e+ 2 2.44e+ 3 0.999 1500 1386. 1428.
12 mean_PPD 2.59e+ 8 1.59e+ 6 1.81e+ 9 1.00 1500 1529. 1236.
13 log-post… 6.32e-86 1.58e-93 1.70e-84 1.00 1500 1413. 1383.# A tibble: 1,500 × 20
.chain .iteration .draw `(Intercept)` `scale(log(DIST))` `scale(log(LDIST))`
<int> <int> <int> <dbl> <dbl> <dbl>
1 1 1 1 2.83 -0.0313 0.133
2 1 2 2 3.14 0.0623 -0.00792
3 1 3 3 3.04 0.0339 0.00387
4 1 4 4 3.05 0.0593 0.0970
5 1 5 5 3.25 0.0645 0.115
6 1 6 6 3.10 0.0190 -0.0320
7 1 7 7 2.97 -0.0420 -0.0309
8 1 8 8 3.06 -0.00284 0.0884
9 1 9 9 2.79 0.0817 -0.00417
10 1 10 10 3.02 0.102 0.0491
# ℹ 1,490 more rows
# ℹ 14 more variables: `scale(log(AREA))` <dbl>, fGRAZE2 <dbl>, fGRAZE3 <dbl>,
# fGRAZE4 <dbl>, fGRAZE5 <dbl>, `scale(ALT)` <dbl>, `scale(YR.ISOL)` <dbl>,
# sigma <dbl>, accept_stat__ <dbl>, stepsize__ <dbl>, treedepth__ <dbl>,
# n_leapfrog__ <dbl>, divergent__ <dbl>, energy__ <dbl># A draws_df: 500 iterations, 3 chains, and 13 variables
(Intercept) scale(log(DIST)) scale(log(LDIST)) scale(log(AREA)) fGRAZE2
1 2.8 -0.0313 0.1326 0.276 0.214
2 3.1 0.0623 -0.0079 0.182 -0.146
3 3.0 0.0339 0.0039 0.260 0.185
4 3.1 0.0593 0.0970 0.123 -0.031
5 3.2 0.0645 0.1151 0.095 0.059
6 3.1 0.0190 -0.0320 0.242 0.113
7 3.0 -0.0420 -0.0309 0.354 0.102
8 3.1 -0.0028 0.0884 0.176 0.129
9 2.8 0.0817 -0.0042 0.372 0.310
10 3.0 0.1020 0.0491 0.061 -0.021
fGRAZE3 fGRAZE4 fGRAZE5
1 0.238 0.328 -0.44
2 -0.038 0.073 -1.14
3 0.047 -0.069 -1.09
4 0.096 -0.040 -0.63
5 -0.286 -0.138 -1.51
6 0.010 0.076 -1.29
7 0.088 0.081 -0.62
8 0.113 0.089 -1.27
9 0.230 0.160 -0.56
10 -0.035 -0.122 -0.69
# ... with 1490 more draws, and 5 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}loyn.rstanarm3$stanfit |>
as_draws_df() |>
summarise_draws(
"median",
~ HDInterval::hdi(.x),
"rhat",
"ess_bulk"
)# A tibble: 13 × 6
variable median lower upper rhat ess_bulk
<chr> <num> <num> <num> <num> <num>
1 (Intercept) 3.05 2.84 3.29 1.00 1323.
2 scale(log(DIST)) -0.0125 -0.122 0.108 1.00 1288.
3 scale(log(LDIST)) 0.0512 -0.0802 0.161 1.00 1482.
4 scale(log(AREA)) 0.213 0.0947 0.333 0.999 1518.
5 fGRAZE2 0.0361 -0.280 0.337 1.00 1546.
6 fGRAZE3 0.0378 -0.207 0.330 1.00 1246.
7 fGRAZE4 0.0110 -0.314 0.322 1.00 1417.
8 fGRAZE5 -1.06 -1.79 -0.438 1.00 1494.
9 scale(ALT) -0.00200 -0.103 0.104 1.00 1384.
10 scale(YR.ISOL) 0.000191 -0.155 0.157 1.00 1485.
11 sigma 6.49 5.37 8.02 0.999 1386.
12 mean_PPD 19.4 16.7 21.5 1.00 1529.
13 log-posterior -196. -202. -192. 1.00 1413.tidyMCMC(loyn.rstanarm3$stanfit, estimate.method='median', conf.int=TRUE,
conf.method='HPDinterval', rhat=TRUE, ess=TRUE)# A tibble: 13 × 7
term estimate std.error conf.low conf.high rhat ess
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 (Intercept) 3.05 0.118 2.84 3.29 1.00 1305
2 scale(log(DIST)) -0.0125 0.0593 -0.122 0.108 1.00 1288
3 scale(log(LDIST)) 0.0512 0.0610 -0.0802 0.161 1.00 1497
4 scale(log(AREA)) 0.213 0.0625 0.0947 0.333 0.999 1502
5 fGRAZE2 0.0361 0.159 -0.280 0.337 1.00 1507
6 fGRAZE3 0.0378 0.138 -0.207 0.330 1.00 1255
7 fGRAZE4 0.0110 0.160 -0.314 0.322 1.00 1434
8 fGRAZE5 -1.06 0.353 -1.79 -0.438 0.999 1477
9 scale(ALT) -0.00200 0.0530 -0.103 0.104 0.999 1374
10 scale(YR.ISOL) 0.000191 0.0794 -0.155 0.157 1.00 1497
11 sigma 6.49 0.696 5.37 8.02 0.999 1377
12 mean_PPD 19.4 1.24 16.7 21.5 1.00 1531
13 log-posterior -196. 2.78 -202. -192. 1.00 1412Conclusions:
- the estimated intercept (expected bird abundance when grazing intensity is equal to 1 and all of the continuous predictors are at their respective averages) is NA. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes NA.
- the estimated slope associated with
DIST(rate at which the abundance of birds changes per 1 unit change in log-transformed Distance to nearest patch holding other continuous predictors constant and grazing intensity at level 1), is NA (mean) or -0.12 with a standard deviation of -0.01. The 90% credibility intervals indicate that we are 90% confident that the slope is between NA and 0.11 - e.g. there is no evidence of a significant trend.
- associated with each of the continuous predictors is a partial slope. Each partial slope is the rate of change between bird abundance and the associated predictor (on the log scale due to the link and based on 1 unit change in the predictor on the scale of the predictor). For example, for every one unit change in centred log patch Area, bird abundance is expected to increase by (log) NA.
- if we back transform this slope (by exponentiation), we get a partial slope for centred log Area of NA. This is interpreted as - for every 1 unit increase in (scaled log) Area, the bird abundance is expected to increase NA fold. That is, there is a NA percent increase per 1 unit increase in centred log Area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
[1] "(Intercept)" "scale(log(DIST))" "scale(log(LDIST))"
[4] "scale(log(AREA))" "fGRAZE2" "fGRAZE3"
[7] "fGRAZE4" "fGRAZE5" "scale(ALT)"
[10] "scale(YR.ISOL)" "sigma" "accept_stat__"
[13] "stepsize__" "treedepth__" "n_leapfrog__"
[16] "divergent__" "energy__" loyn.draw <- loyn.rstanarm3 |> gather_draws(`.Intercept.*|.*AREA.*|.*DIST.*|.*GRAZE.*|.*ALT.*|.*YR.*`, regex=TRUE)
loyn.draw# A tibble: 15,000 × 5
# Groups: .variable [10]
.chain .iteration .draw .variable .value
<int> <int> <int> <chr> <dbl>
1 1 1 1 (Intercept) 2.83
2 1 2 2 (Intercept) 3.14
3 1 3 3 (Intercept) 3.04
4 1 4 4 (Intercept) 3.05
5 1 5 5 (Intercept) 3.25
6 1 6 6 (Intercept) 3.10
7 1 7 7 (Intercept) 2.97
8 1 8 8 (Intercept) 3.06
9 1 9 9 (Intercept) 2.79
10 1 10 10 (Intercept) 3.02
# ℹ 14,990 more rows
loyn.rstanarm3 |>
gather_draws(`.*AREA.*|.*DIST.*|.*GRAZE.*|.*ALT.*|.*YR.*`, regex=TRUE) |>
ggplot() +
stat_halfeye(aes(x=.value, y=.variable)) +
facet_wrap(~.variable, scales='free')
loyn.rstanarm3 |>
gather_draws(`.*AREA.*|.*DIST.*|.*GRAZE.*|.*ALT.*|.*YR.*`, regex=TRUE) |>
ggplot() +
stat_halfeye(aes(x=.value, y=.variable)) +
geom_vline(xintercept = 1, linetype = 'dashed')
loyn.rstanarm3 |>
gather_draws(`.*AREA.*|.*DIST.*|.*GRAZE.*|.*ALT.*|.*YR.*`, regex=TRUE) |>
ggplot() +
geom_density_ridges(aes(x=.value, y = .variable), alpha=0.4) +
geom_vline(xintercept = 0, linetype = 'dashed')Picking joint bandwidth of 0.0251
##Or on a fractional scale
loyn.rstanarm3 |>
gather_draws(`.*AREA.*|.*DIST.*|.*GRAZE.*|.*ALT.*|.*YR.*`, regex=TRUE) |>
ggplot() +
geom_density_ridges_gradient(aes(x=exp(.value),
y = .variable,
fill = stat(x)),
alpha=0.4, colour = 'white',
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)) +
geom_vline(xintercept = 1, linetype = 'dashed') +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_c(option = "C")Warning: `stat(x)` was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(x)` instead.Picking joint bandwidth of 0.0363Warning: Using the `size` aesthetic with geom_segment was deprecated in ggplot2 3.4.0.
ℹ Please use the `linewidth` aesthetic instead.
loyn.rstanarm3 |> spread_draws(`.Intercept.*|.*DIST.*|.*AREA.*|.*GRAZE.*|.*ALT.*|.*YR.*`, regex=TRUE) # A tibble: 1,500 × 13
.chain .iteration .draw `(Intercept)` `scale(log(DIST))` `scale(log(LDIST))`
<int> <int> <int> <dbl> <dbl> <dbl>
1 1 1 1 2.83 -0.0313 0.133
2 1 2 2 3.14 0.0623 -0.00792
3 1 3 3 3.04 0.0339 0.00387
4 1 4 4 3.05 0.0593 0.0970
5 1 5 5 3.25 0.0645 0.115
6 1 6 6 3.10 0.0190 -0.0320
7 1 7 7 2.97 -0.0420 -0.0309
8 1 8 8 3.06 -0.00284 0.0884
9 1 9 9 2.79 0.0817 -0.00417
10 1 10 10 3.02 0.102 0.0491
# ℹ 1,490 more rows
# ℹ 7 more variables: `scale(log(AREA))` <dbl>, fGRAZE2 <dbl>, fGRAZE3 <dbl>,
# fGRAZE4 <dbl>, fGRAZE5 <dbl>, `scale(ALT)` <dbl>, `scale(YR.ISOL)` <dbl>Warning: `as.tibble()` was deprecated in tibble 2.0.0.
ℹ Please use `as_tibble()` instead.
ℹ The signature and semantics have changed, see `?as_tibble`.Warning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.# A tibble: 1,500 × 11
`(Intercept)` `scale(log(DIST))` `scale(log(LDIST))` `scale(log(AREA))`
<dbl> <dbl> <dbl> <dbl>
1 2.83 -0.0313 0.133 0.276
2 3.14 0.0623 -0.00792 0.182
3 3.04 0.0339 0.00387 0.260
4 3.05 0.0593 0.0970 0.123
5 3.25 0.0645 0.115 0.0953
6 3.10 0.0190 -0.0320 0.242
7 2.97 -0.0420 -0.0309 0.354
8 3.06 -0.00284 0.0884 0.176
9 2.79 0.0817 -0.00417 0.372
10 3.02 0.102 0.0491 0.0613
# ℹ 1,490 more rows
# ℹ 7 more variables: fGRAZE2 <dbl>, fGRAZE3 <dbl>, fGRAZE4 <dbl>,
# fGRAZE5 <dbl>, `scale(ALT)` <dbl>, `scale(YR.ISOL)` <dbl>, sigma <dbl>The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).
Family: gaussian
Links: mu = log; sigma = identity
Formula: ABUND ~ scale(log(DIST)) + scale(log(LDIST)) + scale(log(AREA)) + fGRAZE + scale(ALT) + scale(YR.ISOL)
Data: loyn (Number of observations: 56)
Draws: 3 chains, each with iter = 5000; warmup = 2500; thin = 5;
total post-warmup draws = 1500
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 3.06 0.12 2.82 3.29 1.00 1547 1495
scalelogDIST -0.01 0.06 -0.13 0.10 1.00 1259 1417
scalelogLDIST 0.05 0.06 -0.06 0.18 1.00 1406 1459
scalelogAREA 0.21 0.06 0.08 0.32 1.00 1521 1420
fGRAZE2 0.03 0.16 -0.28 0.32 1.00 1519 1347
fGRAZE3 0.03 0.14 -0.26 0.31 1.00 1573 1417
fGRAZE4 -0.01 0.17 -0.34 0.33 1.00 1664 1501
fGRAZE5 -1.13 0.37 -1.90 -0.49 1.00 1476 1343
scaleALT 0.00 0.05 -0.10 0.11 1.00 1345 1537
scaleYR.ISOL -0.00 0.08 -0.15 0.16 1.00 1285 1349
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.50 0.69 5.35 7.99 1.00 1448 1499
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Conclusions:
- in the Model Info, we are informed that the total MCMC posterior sample size is 1500 and that there were 56 raw observations.
- the estimated intercept (expected bird abundance when grazing intensity is equal to 1 and all of the continuous predictors are at their respective averages) is 3.06. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes 21.32.
- the estimated slope associated with
DIST(rate at which the abundance of birds changes per 1 unit change in log-transformed Distance to nearest patch holding other continuous predictors constant and grazing intensity at level 1), is -0.01 (mean) or 0.1 (median) with a standard deviation of 0.06. The 90% credibility intervals indicate that we are 90% confident that the slope is between -0.01 and 1 - e.g. there is no evidence of a significant trend.
- associated with each of the continuous predictors is a partial slope. Each partial slope is the rate of change between bird abundance and the associated predictor (on the log scale due to the link and based on 1 unit change in the predictor on the scale of the predictor). For example, for every one unit change in centred log patch Area, bird abundance is expected to increase by (log) 0.21.
- if we back transform this slope (by exponentiation), we get a partial slope for centred log Area of 1.23. This is interpreted as - for every 1 unit increase in (scaled log) Area, the bird abundance is expected to increase 1.23 fold. That is, there is a 22.77 percent increase per 1 unit increase in centred log Area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
# A tibble: 1,500 × 25
.chain .iteration .draw b_Intercept b_scalelogDIST b_scalelogLDIST
<int> <int> <int> <dbl> <dbl> <dbl>
1 1 1 1 3.02 0.0116 0.0635
2 1 2 2 3.08 0.0222 0.125
3 1 3 3 3.08 -0.0793 0.0630
4 1 4 4 3.19 0.0109 0.141
5 1 5 5 2.90 0.0193 0.00407
6 1 6 6 3.14 0.0468 0.0406
7 1 7 7 2.93 -0.0550 0.121
8 1 8 8 2.79 -0.0622 0.176
9 1 9 9 3.21 -0.101 0.121
10 1 10 10 3.15 -0.0297 0.0896
# ℹ 1,490 more rows
# ℹ 19 more variables: b_scalelogAREA <dbl>, b_fGRAZE2 <dbl>, b_fGRAZE3 <dbl>,
# b_fGRAZE4 <dbl>, b_fGRAZE5 <dbl>, b_scaleALT <dbl>, b_scaleYR.ISOL <dbl>,
# sigma <dbl>, prior_Intercept <dbl>, prior_b <dbl>, prior_sigma <dbl>,
# lprior <dbl>, lp__ <dbl>, accept_stat__ <dbl>, treedepth__ <dbl>,
# stepsize__ <dbl>, divergent__ <dbl>, n_leapfrog__ <dbl>, energy__ <dbl>loyn.brm3 |>
tidy_draws() |>
dplyr::select(starts_with("b_")) |>
exp() |>
summarise_draws(median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail)# A tibble: 10 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <num> <num> <num> <num> <num> <num> <num>
1 b_Intercept 21.4 16.5 26.3 1.00 1500 1535. 1459.
2 b_scalelogDIST 0.990 0.877 1.11 1.00 1500 1247. 1395.
3 b_scalelogLDIST 1.05 0.938 1.19 0.999 1500 1365. 1447.
4 b_scalelogAREA 1.23 1.09 1.38 1.00 1500 1509. 1414.
5 b_fGRAZE2 1.04 0.713 1.33 1.00 1500 1488. 1340.
6 b_fGRAZE3 1.03 0.755 1.34 0.999 1500 1564. 1414.
7 b_fGRAZE4 0.996 0.669 1.31 1.00 1500 1595. 1457.
8 b_fGRAZE5 0.330 0.121 0.566 1.00 1500 1462. 1328.
9 b_scaleALT 1.00 0.898 1.11 1.00 1500 1333. 1512.
10 b_scaleYR.ISOL 0.995 0.852 1.15 1.00 1500 1261. 1334.loyn.brm3 |>
spread_draws(`^b_.*|sigma`, regex = TRUE) |>
exp() |>
summarise_draws(median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail)# A tibble: 11 × 8
variable median lower upper rhat length ess_bulk ess_tail
<chr> <num> <num> <num> <num> <num> <num> <num>
1 b_Intercept 21.4 16.5 26.3 1.00 1500 1547. 1495.
2 b_scalelogDIST 0.990 0.877 1.11 1.00 1500 1260. 1417.
3 b_scalelogLDIST 1.05 0.938 1.19 1.00 1500 1406. 1459.
4 b_scalelogAREA 1.23 1.09 1.38 1.00 1500 1521. 1420.
5 b_fGRAZE2 1.04 0.713 1.33 1.00 1500 1518. 1347.
6 b_fGRAZE3 1.03 0.755 1.34 0.999 1500 1573. 1417.
7 b_fGRAZE4 0.996 0.669 1.31 1.00 1500 1664. 1501.
8 b_fGRAZE5 0.330 0.121 0.566 1.00 1500 1476. 1343.
9 b_scaleALT 1.00 0.898 1.11 1.00 1500 1345. 1537.
10 b_scaleYR.ISOL 0.995 0.852 1.15 1.00 1500 1284. 1349.
11 sigma 618. 126. 2284. 1.00 1500 1448. 1499.# A draws_df: 500 iterations, 3 chains, and 16 variables
b_Intercept b_scalelogDIST b_scalelogLDIST b_scalelogAREA b_fGRAZE2
1 3.0 0.012 0.0635 0.18 0.042
2 3.1 0.022 0.1248 0.17 -0.127
3 3.1 -0.079 0.0630 0.26 -0.025
4 3.2 0.011 0.1411 0.14 -0.098
5 2.9 0.019 0.0041 0.24 0.286
6 3.1 0.047 0.0406 0.18 0.100
7 2.9 -0.055 0.1210 0.33 0.013
8 2.8 -0.062 0.1760 0.20 -0.040
9 3.2 -0.101 0.1209 0.17 -0.110
10 3.1 -0.030 0.0896 0.21 0.034
b_fGRAZE3 b_fGRAZE4 b_fGRAZE5
1 0.0105 -0.0082 -0.86
2 0.0084 -0.0057 -0.93
3 0.0361 -0.0822 -0.91
4 -0.0507 -0.0611 -0.96
5 0.1847 0.1297 -0.94
6 -0.0695 -0.0797 -1.04
7 0.0816 0.1376 -0.94
8 0.3040 0.0972 -0.27
9 -0.2545 -0.1272 -1.90
10 -0.0864 0.1405 -1.27
# ... with 1490 more draws, and 8 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}loyn.brm3 |>
as_draws_df() |>
summarise_draws(
"median",
~ HDInterval::hdi(.x),
"rhat",
"ess_bulk"
)# A tibble: 16 × 6
variable median lower upper rhat ess_bulk
<chr> <num> <num> <num> <num> <num>
1 b_Intercept 3.06 2.82 3.29 1.00 1547.
2 b_scalelogDIST -0.0102 -0.129 0.106 1.00 1260.
3 b_scalelogLDIST 0.0533 -0.0645 0.177 1.00 1406.
4 b_scalelogAREA 0.206 0.0879 0.323 1.00 1521.
5 b_fGRAZE2 0.0377 -0.288 0.322 1.00 1518.
6 b_fGRAZE3 0.0255 -0.232 0.326 0.999 1573.
7 b_fGRAZE4 -0.00450 -0.368 0.293 1.00 1664.
8 b_fGRAZE5 -1.11 -1.86 -0.466 1.00 1476.
9 b_scaleALT 0.00165 -0.107 0.104 1.00 1345.
10 b_scaleYR.ISOL -0.00543 -0.160 0.143 1.00 1284.
11 sigma 6.43 5.25 7.81 1.00 1448.
12 prior_Intercept 2.98 1.97 3.92 1.00 1321.
13 prior_b -0.000828 -3.08 2.79 1.00 1573.
14 prior_sigma 1.95 0.000907 7.76 0.999 1483.
15 lprior -16.1 -16.9 -15.5 1.00 1398.
16 lp__ -199. -204. -194. 0.999 1516.loyn.brm3$fit |>
tidyMCMC(estimate.method = 'median',
conf.int = TRUE, conf.method = 'HPDinterval',
rhat = TRUE, ess = TRUE)# A tibble: 15 × 7
term estimate std.error conf.low conf.high rhat ess
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 b_Intercept 3.06 0.117 2.82 3.29 1.00 1512
2 b_scalelogDIST -0.0102 0.0602 -0.129 0.106 1.00 1257
3 b_scalelogLDIST 0.0533 0.0616 -0.0645 0.177 0.999 1374
4 b_scalelogAREA 0.206 0.0592 0.0879 0.323 0.999 1506
5 b_fGRAZE2 0.0377 0.156 -0.288 0.322 1.00 1499
6 b_fGRAZE3 0.0255 0.143 -0.232 0.326 0.999 1567
7 b_fGRAZE4 -0.00450 0.167 -0.368 0.293 1.00 1659
8 b_fGRAZE5 -1.11 0.365 -1.86 -0.466 1.00 1454
9 b_scaleALT 0.00165 0.0527 -0.107 0.104 1.00 1335
10 b_scaleYR.ISOL -0.00543 0.0780 -0.160 0.143 1.00 1243
11 sigma 6.43 0.690 5.25 7.81 0.999 1440
12 prior_Intercept 2.98 0.501 1.97 3.92 1.00 1316
13 prior_b -0.000828 1.51 -3.08 2.79 0.999 1552
14 prior_sigma 1.95 2.93 0.000907 7.76 0.998 1470
15 lprior -16.1 0.401 -16.9 -15.5 0.998 1348Conclusions:
- the estimated intercept (expected bird abundance when grazing intensity is equal to 1 and all of the continuous predictors are at their respective averages) is NA. This is the mean of the posterior distribution for this parameter. If we back-transform this to the response scale, this becomes NA.
- the estimated slope associated with
DIST(rate at which the abundance of birds changes per 1 unit change in log-transformed Distance to nearest patch holding other continuous predictors constant and grazing intensity at level 1), is NA (mean) or -0.13 with a standard deviation of -0.01. The 90% credibility intervals indicate that we are 90% confident that the slope is between NA and 0.11 - e.g. there is no evidence of a significant trend.
- associated with each of the continuous predictors is a partial slope. Each partial slope is the rate of change between bird abundance and the associated predictor (on the log scale due to the link and based on 1 unit change in the predictor on the scale of the predictor). For example, for every one unit change in centred log patch Area, bird abundance is expected to increase by (log) NA.
- if we back transform this slope (by exponentiation), we get a partial slope for centred log Area of NA. This is interpreted as - for every 1 unit increase in (scaled log) Area, the bird abundance is expected to increase NA fold. That is, there is a NA percent increase per 1 unit increase in centred log Area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
[1] "b_Intercept" "b_scalelogDIST" "b_scalelogLDIST" "b_scalelogAREA"
[5] "b_fGRAZE2" "b_fGRAZE3" "b_fGRAZE4" "b_fGRAZE5"
[9] "b_scaleALT" "b_scaleYR.ISOL" "sigma" "prior_Intercept"
[13] "prior_b" "prior_sigma" "lprior" "lp__"
[17] "accept_stat__" "treedepth__" "stepsize__" "divergent__"
[21] "n_leapfrog__" "energy__" # A tibble: 15,000 × 5
# Groups: .variable [10]
.chain .iteration .draw .variable .value
<int> <int> <int> <chr> <dbl>
1 1 1 1 b_Intercept 3.02
2 1 2 2 b_Intercept 3.08
3 1 3 3 b_Intercept 3.08
4 1 4 4 b_Intercept 3.19
5 1 5 5 b_Intercept 2.90
6 1 6 6 b_Intercept 3.14
7 1 7 7 b_Intercept 2.93
8 1 8 8 b_Intercept 2.79
9 1 9 9 b_Intercept 3.21
10 1 10 10 b_Intercept 3.15
# ℹ 14,990 more rows
loyn.brm3 |>
gather_draws(`^b_.*`, regex=TRUE) |>
filter(.variable != 'b_Intercept') |>
ggplot() +
stat_halfeye(aes(x=.value, y=.variable)) +
facet_wrap(~.variable, scales='free')
loyn.brm3 |>
gather_draws(`^b_.*`, regex=TRUE) |>
filter(.variable != 'b_Intercept') |>
ggplot() +
stat_halfeye(aes(x=.value, y=.variable)) +
geom_vline(xintercept = 0, linetype = 'dashed')
loyn.brm3 |>
gather_draws(`^b_.*`, regex=TRUE) |>
filter(.variable != 'b_Intercept') |>
ggplot() +
geom_density_ridges(aes(x=.value, y = .variable), alpha=0.4) +
geom_vline(xintercept = 0, linetype = 'dashed')Picking joint bandwidth of 0.0258
##Or on a fractional scale
loyn.brm3 |>
gather_draws(`^b_.*`, regex=TRUE) |>
filter(.variable != 'b_Intercept') |>
ggplot() +
geom_density_ridges_gradient(aes(x=exp(.value),
y = .variable,
fill = stat(x)),
alpha=0.4, colour = 'white',
quantile_lines = TRUE,
quantiles = c(0.025, 0.975)) +
geom_vline(xintercept = 1, linetype = 'dashed') +
scale_x_continuous(trans = scales::log2_trans()) +
scale_fill_viridis_c(option = "C")Picking joint bandwidth of 0.0373
# A tibble: 1,500 × 13
.chain .iteration .draw b_Intercept b_scalelogDIST b_scalelogLDIST
<int> <int> <int> <dbl> <dbl> <dbl>
1 1 1 1 3.02 0.0116 0.0635
2 1 2 2 3.08 0.0222 0.125
3 1 3 3 3.08 -0.0793 0.0630
4 1 4 4 3.19 0.0109 0.141
5 1 5 5 2.90 0.0193 0.00407
6 1 6 6 3.14 0.0468 0.0406
7 1 7 7 2.93 -0.0550 0.121
8 1 8 8 2.79 -0.0622 0.176
9 1 9 9 3.21 -0.101 0.121
10 1 10 10 3.15 -0.0297 0.0896
# ℹ 1,490 more rows
# ℹ 7 more variables: b_scalelogAREA <dbl>, b_fGRAZE2 <dbl>, b_fGRAZE3 <dbl>,
# b_fGRAZE4 <dbl>, b_fGRAZE5 <dbl>, b_scaleALT <dbl>, b_scaleYR.ISOL <dbl>Warning: Method 'posterior_samples' is deprecated. Please see ?as_draws for
recommended alternatives.# A tibble: 1,500 × 16
b_Intercept b_scalelogDIST b_scalelogLDIST b_scalelogAREA b_fGRAZE2 b_fGRAZE3
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 3.02 0.0116 0.0635 0.184 0.0421 0.0105
2 3.08 0.0222 0.125 0.174 -0.127 0.00843
3 3.08 -0.0793 0.0630 0.261 -0.0252 0.0361
4 3.19 0.0109 0.141 0.140 -0.0984 -0.0507
5 2.90 0.0193 0.00407 0.241 0.286 0.185
6 3.14 0.0468 0.0406 0.184 0.100 -0.0695
7 2.93 -0.0550 0.121 0.333 0.0133 0.0816
8 2.79 -0.0622 0.176 0.200 -0.0396 0.304
9 3.21 -0.101 0.121 0.166 -0.110 -0.255
10 3.15 -0.0297 0.0896 0.211 0.0343 -0.0864
# ℹ 1,490 more rows
# ℹ 10 more variables: b_fGRAZE4 <dbl>, b_fGRAZE5 <dbl>, b_scaleALT <dbl>,
# b_scaleYR.ISOL <dbl>, sigma <dbl>, prior_Intercept <dbl>, prior_b <dbl>,
# prior_sigma <dbl>, lprior <dbl>, lp__ <dbl>Region of Practical Equivalence
For standardised parameter (negligible effect)
[1] 0.09024351[1] -0.01 0.01# Proportion of samples inside the ROPE [-0.09, 0.09]:
Parameter | inside ROPE
---------------------------
Intercept | 0.00 %
scalelogDIST | 91.78 %
scalelogLDIST | 73.67 %
scalelogAREA | 0.35 %
fGRAZE2 | 43.75 %
fGRAZE3 | 49.37 %
fGRAZE4 | 44.52 %
fGRAZE5 | 0.00 %
scaleALT | 96.07 %
scaleYR.ISOL | 79.85 %
10 Further analyses
loyn.rstanarm4a <- update(loyn.rstanarm3, .~scale(log(DIST))*scale(log(LDIST)),
diagnostic_file = file.path(tempdir(), "dfa.csv"))
loyn.rstanarm4b <- update(loyn.rstanarm3, .~scale(log(AREA)) * fGRAZE,
diagnostic_file = file.path(tempdir(), "dfb.csv"))
loyn.rstanarm4c <- update(loyn.rstanarm3, .~scale(log(AREA)) * fGRAZE * scale(YR.ISOL),
diagnostic_file = file.path(tempdir(), "dfc.csv"))
loyn.rstanarm4d <- update(loyn.rstanarm3, .~scale(ALT),
diagnostic_file = file.path(tempdir(), "dfd.csv"))
loyn.rstanarm4e <- update(loyn.rstanarm3, .~1,
diagnostic_file = file.path(tempdir(), "dfe.csv"))
loo_compare(loo(loyn.rstanarm4a),
loo(loyn.rstanarm4e)
)Warning: Found 1 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 1 times to compute the ELPDs for the problematic observations directly. elpd_diff se_diff
loyn.rstanarm4e 0.0 0.0
loyn.rstanarm4a -2.5 1.6 elpd_diff se_diff
loyn.rstanarm4b 0.0 0.0
loyn.rstanarm4e -30.6 7.2 Warning: Found 5 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 5 times to compute the ELPDs for the problematic observations directly. elpd_diff se_diff
loyn.rstanarm4c 0.0 0.0
loyn.rstanarm4e -22.3 6.7 elpd_diff se_diff
loyn.rstanarm4d 0.0 0.0
loyn.rstanarm4e -3.5 2.3 Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6
Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5Estimated Bayes factor in favor of x1 over x2: 0.00139Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6
Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5Estimated Bayes factor in favor of x1 over x2: 1081033284.51737Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6
Iteration: 7
Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5Estimated Bayes factor in favor of x1 over x2: 7540.31759Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6Estimated Bayes factor in favor of x1 over x2: 4.97595loyn.brm4a <- update(loyn.brm3, .~scale(log(DIST))*scale(log(LDIST)),
save_pars = save_pars(all = TRUE), refresh = 0)Start samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 0.2 seconds.
Chain 2 finished in 0.2 seconds.
Chain 3 finished in 0.2 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.2 seconds.
Total execution time: 0.7 seconds.loyn.brm4b <- update(loyn.brm3, .~scale(log(AREA)) * fGRAZE,
save_pars = save_pars(all = TRUE), refresh = 0)Start samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 0.3 seconds.
Chain 2 finished in 0.3 seconds.
Chain 3 finished in 0.3 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.3 seconds.
Total execution time: 1.3 seconds.loyn.brm4c <- update(loyn.brm3, .~scale(log(AREA)) * fGRAZE * scale(YR.ISOL),
save_pars = save_pars(all = TRUE), refresh = 0)Start samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 1.4 seconds.
Chain 2 finished in 1.5 seconds.
Chain 3 finished in 1.7 seconds.
All 3 chains finished successfully.
Mean chain execution time: 1.5 seconds.
Total execution time: 4.9 seconds.Start samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 0.1 seconds.
Chain 2 finished in 0.1 seconds.
Chain 3 finished in 0.1 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.1 seconds.
Total execution time: 0.8 seconds.The desired updates require recompiling the model
Start samplingRunning MCMC with 3 sequential chains...
Chain 1 finished in 0.1 seconds.
Chain 2 finished in 0.1 seconds.
Chain 3 finished in 0.1 seconds.
All 3 chains finished successfully.
Mean chain execution time: 0.1 seconds.
Total execution time: 0.4 seconds.Warning:
1 (1.8%) p_waic estimates greater than 0.4. We recommend trying loo instead.
Computed from 1500 by 56 log-likelihood matrix
Estimate SE
elpd_waic -215.9 3.6
p_waic 4.5 0.8
waic 431.8 7.2
1 (1.8%) p_waic estimates greater than 0.4. We recommend trying loo instead.
Computed from 1500 by 56 log-likelihood matrix
Estimate SE
elpd_loo -216.0 3.6
p_loo 4.6 0.8
looic 432.0 7.2
------
Monte Carlo SE of elpd_loo is 0.1.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 55 98.2% 674
(0.5, 0.7] (ok) 1 1.8% 624
(0.7, 1] (bad) 0 0.0% <NA>
(1, Inf) (very bad) 0 0.0% <NA>
All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.
Computed from 1500 by 56 log-likelihood matrix
Estimate SE
elpd_loo -213.7 3.9
p_loo 1.6 0.2
looic 427.4 7.7
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details. elpd_diff se_diff
loyn.brm4e 0.0 0.0
loyn.brm4a -2.3 1.6 Warning: Found 3 observations with a pareto_k > 0.7 in model 'loyn.brm4b'. It
is recommended to set 'moment_match = TRUE' in order to perform moment matching
for problematic observations. elpd_diff se_diff
loyn.brm4b 0.0 0.0
loyn.brm4e -30.4 7.4 Recompiling the model with 'rstan'Recompilation doneWarning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details. elpd_diff se_diff
loyn.brm4b 0.0 0.0
loyn.brm4e -30.4 7.4 Recompiling the model with 'rstan'
Recompilation doneWarning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details. elpd_diff se_diff
loyn.brm4c 0.0 0.0
loyn.brm4e -21.7 6.9 elpd_diff se_diff
loyn.brm4d 0.0 0.0
loyn.brm4e -3.6 2.3 An alternative is to compute Bayes factors based on bridge sampling. Note, this process usually requires far greater number of posterior samples (10x) in order to be stable. It is also advisable to run this multiple times to ensure stability.
It calculates the marginal likelihood of one model in favour of another. The larger the value, the more evidence there is of one model over the other.
Recompiling the model with 'rstan'Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6Estimated Bayes factor in favor of loyn.brm4a over loyn.brm4e: 0.00040Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4Estimated Bayes factor in favor of loyn.brm4e over loyn.brm4a: 2433.63374Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5Estimated Bayes factor in favor of loyn.brm4b over loyn.brm4e: 116156587.39543Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Iteration: 5
Iteration: 6
Iteration: 7Estimated Bayes factor in favor of loyn.brm4e over loyn.brm4b: 0.00000Recompiling the model with 'rstan'
Recompilation doneIteration: 1
Iteration: 2
Iteration: 3
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Iteration: 8Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 6Estimated Bayes factor in favor of loyn.brm4c over loyn.brm4e: 50.56553Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 5Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 8Estimated Bayes factor in favor of loyn.brm4e over loyn.brm4c: 0.02034Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 5Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 5Estimated Bayes factor in favor of loyn.brm4d over loyn.brm4e: 3.76348Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 5Recompiling the model with 'rstan'
Recompilation doneIteration: 1
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Iteration: 6Estimated Bayes factor in favor of loyn.brm4e over loyn.brm4d: 0.26484Compare effect of grazing at different patch areas
loyn.list <- with(loyn, list(AREA = c(min(AREA), mean(AREA), max(AREA))))
loyn.brm4b |>
emmeans(~fGRAZE|AREA, at = loyn.list, type = "response") |>
pairs(reverse = FALSE)AREA = 0.1:
contrast ratio lower.HPD upper.HPD
fGRAZE1 / fGRAZE2 2.027 0.87934 4.122
fGRAZE1 / fGRAZE3 2.234 0.99716 4.190
fGRAZE1 / fGRAZE4 9.429 0.96176 54.816
fGRAZE1 / fGRAZE5 11.269 0.88471 70.297
fGRAZE2 / fGRAZE3 1.111 0.29762 2.204
fGRAZE2 / fGRAZE4 4.576 0.25708 27.668
fGRAZE2 / fGRAZE5 5.518 0.37513 34.499
fGRAZE3 / fGRAZE4 4.179 0.30209 23.917
fGRAZE3 / fGRAZE5 5.062 0.43230 31.686
fGRAZE4 / fGRAZE5 1.216 0.02015 10.576
AREA = 69.2696428571429:
contrast ratio lower.HPD upper.HPD
fGRAZE1 / fGRAZE2 0.905 0.69062 1.144
fGRAZE1 / fGRAZE3 0.848 0.64652 1.122
fGRAZE1 / fGRAZE4 0.494 0.20616 0.930
fGRAZE1 / fGRAZE5 1.484 0.36803 3.921
fGRAZE2 / fGRAZE3 0.937 0.64195 1.285
fGRAZE2 / fGRAZE4 0.541 0.21900 1.041
fGRAZE2 / fGRAZE5 1.629 0.35264 4.347
fGRAZE3 / fGRAZE4 0.582 0.22298 1.116
fGRAZE3 / fGRAZE5 1.741 0.35246 4.604
fGRAZE4 / fGRAZE5 3.082 0.60455 9.184
AREA = 1771:
contrast ratio lower.HPD upper.HPD
fGRAZE1 / fGRAZE2 0.597 0.29537 0.981
fGRAZE1 / fGRAZE3 0.521 0.25267 0.921
fGRAZE1 / fGRAZE4 0.113 0.00336 0.524
fGRAZE1 / fGRAZE5 0.539 0.00253 3.518
fGRAZE2 / fGRAZE3 0.868 0.29228 1.743
fGRAZE2 / fGRAZE4 0.189 0.00625 0.965
fGRAZE2 / fGRAZE5 0.883 0.00289 6.202
fGRAZE3 / fGRAZE4 0.218 0.00322 1.048
fGRAZE3 / fGRAZE5 1.013 0.00437 7.075
fGRAZE4 / fGRAZE5 4.790 0.00837 56.746
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 newdata <- loyn.brm4b |>
emmeans(~fGRAZE|AREA, at = loyn.list, type = 'response') |>
pairs() |>
as.data.frame()
head(newdata) contrast AREA ratio lower.HPD upper.HPD
fGRAZE1 / fGRAZE2 0.1 2.026957 0.8793432 4.12190
fGRAZE1 / fGRAZE3 0.1 2.234104 0.9971590 4.18955
fGRAZE1 / fGRAZE4 0.1 9.429410 0.9617564 54.81644
fGRAZE1 / fGRAZE5 0.1 11.268673 0.8847058 70.29720
fGRAZE2 / fGRAZE3 0.1 1.110790 0.2976185 2.20414
fGRAZE2 / fGRAZE4 0.1 4.575734 0.2570813 27.66753
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 newdata <- loyn.brm4b |>
emmeans(~fGRAZE|AREA, at = loyn.list, type = 'response') |>
pairs() |>
gather_emmeans_draws()
newdata |> median_hdci() |>
ggplot() +
geom_hline(yintercept = 1, linetype = 'dashed') +
geom_pointrange(aes(y = .value, ymin = .lower, ymax = .upper, x = contrast)) +
facet_wrap(~AREA) +
coord_flip()
# A tibble: 22,500 × 6
# Groups: fGRAZE, AREA [15]
fGRAZE AREA .chain .iteration .draw .value
<fct> <dbl> <int> <int> <int> <dbl>
1 1 0.1 NA NA 1 2.87
2 1 0.1 NA NA 2 2.91
3 1 0.1 NA NA 3 2.52
4 1 0.1 NA NA 4 2.61
5 1 0.1 NA NA 5 2.88
6 1 0.1 NA NA 6 3.12
7 1 0.1 NA NA 7 2.96
8 1 0.1 NA NA 8 2.86
9 1 0.1 NA NA 9 3.32
10 1 0.1 NA NA 10 2.90
# ℹ 22,490 more rows`summarise()` has grouped output by 'contrast'. You can override using the
`.groups` argument.g <- newdata |>
ggplot() +
geom_vline(xintercept = 1, linetype = 'dashed') +
stat_slab(aes(x = .value, y = contrast,
fill = stat(ggdist::cut_cdf_qi(cdf,
.width = c(0.5, 0.8, 0.95),
labels = scales::percent_format())
)), color = 'black') +
scale_fill_brewer('Interval', direction = -1, na.translate = FALSE) +
facet_grid(~round(AREA,1))
11 Summary figure
loyn.list <- with(loyn, list(AREA = modelr::seq_range(AREA, n=100)))
newdata <- emmeans(loyn.brm3, ~AREA|fGRAZE, at = loyn.list, type='response') |>
as.data.frame()
head(newdata) AREA fGRAZE response lower.HPD upper.HPD
0.10000 1 13.35678 8.347263 21.02803
17.98788 1 23.72537 18.959959 28.43506
35.87576 1 25.60432 21.472996 30.50046
53.76364 1 26.77315 22.353425 31.31426
71.65152 1 27.61484 23.291129 32.27396
89.53939 1 28.27732 23.650035 32.87856
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 ggplot(newdata, aes(y=response, x=AREA)) +
geom_ribbon(aes(ymin=lower.HPD, ymax=upper.HPD, fill=fGRAZE), alpha=0.3) +
geom_line(aes(color=fGRAZE)) +
theme_bw() +
scale_x_log10() +
scale_y_log10()
spaghetti = emmeans(loyn.rstanarm3, ~AREA|fGRAZE, at = loyn.list, type='response') |>
gather_emmeans_draws() |>
mutate(Fit=exp(.value))Warning: Model has 0 prior weights, but we recovered 56 rows of data.
So prior weights were ignored.wch = sample(1:max(spaghetti$.draw), 100,replace=FALSE)
spaghetti = spaghetti |>
filter(.draw %in% wch)
ggplot(newdata) +
geom_line(data=spaghetti, aes(y=Fit, x=AREA, color=fGRAZE,
group=interaction(fGRAZE,.draw)), alpha=0.05) +
geom_line(aes(y=response, x=AREA, color=fGRAZE)) +
theme_bw() +
scale_x_log10() + scale_y_log10()
## or honouring the data range
loyn.nd <- loyn |>
group_by(fGRAZE) |>
expand(AREA = modelr::seq_range(AREA, n=100),
DIST = mean(DIST), LDIST = mean(LDIST), ALT = mean(ALT), YR.ISOL = mean(YR.ISOL))
loyn.rstanarm3 |>
epred_draws(newdata = loyn.nd, value = '.value') |>
median_hdci() |>
ggplot(aes(x = AREA, y = .value, colour = fGRAZE, fill = fGRAZE)) +
geom_ribbon(aes(ymin = .lower, ymax = .upper), colour = NA, alpha = 0.3) +
geom_line() +
geom_point(data = loyn, aes(y = ABUND, x = AREA)) +
scale_y_log10() +
scale_x_log10() +
theme_bw()
loyn.list = with(loyn, list(AREA = seq(min(AREA), max(AREA), len=100)))
newdata = emmeans(loyn.rstanarm4b, ~AREA|fGRAZE, at = loyn.list, type='response') |>
as.data.frame()Warning: Model has 0 prior weights, but we recovered 56 rows of data.
So prior weights were ignored. AREA fGRAZE response lower.HPD upper.HPD
0.10000 1 19.33407 11.90402 27.95127
17.98788 1 26.26695 22.54694 30.20361
35.87576 1 27.30720 23.76920 30.55262
53.76364 1 27.95785 24.77001 31.21964
71.65152 1 28.41624 25.34805 31.79729
89.53939 1 28.83454 25.62688 32.11486
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 ggplot(newdata, aes(y=response, x=AREA)) +
geom_ribbon(aes(ymin=lower.HPD, ymax=upper.HPD, fill=fGRAZE), alpha=0.3) +
geom_line(aes(color=fGRAZE)) +
theme_bw() +
scale_x_log10() +
scale_y_log10()
spaghetti = emmeans(loyn.rstanarm4b, ~AREA|fGRAZE, at = loyn.list, type='response') |>
gather_emmeans_draws() |> mutate(Fit=exp(.value))Warning: Model has 0 prior weights, but we recovered 56 rows of data.
So prior weights were ignored.wch = sample(1:max(spaghetti$.draw), 100,replace=FALSE)
spaghetti = spaghetti |> filter(.draw %in% wch)
ggplot(newdata) +
geom_line(data=spaghetti, aes(y=Fit, x=AREA, color=fGRAZE,
group=interaction(fGRAZE,.draw)), alpha=0.05) +
geom_line(aes(y=response, x=AREA, color=fGRAZE)) +
theme_bw() +
scale_x_log10() + scale_y_log10()
## or honouring the data range
loyn.nd <- loyn |>
group_by(fGRAZE) |>
expand(AREA = modelr::seq_range(AREA, n=100))
loyn.rstanarm4b |>
epred_draws(newdata = loyn.nd, value = '.value') |>
median_hdci() |>
ggplot(aes(x = AREA, y = .value, colour = fGRAZE, fill = fGRAZE)) +
geom_ribbon(aes(ymin = .lower, ymax = .upper), colour = NA, alpha = 0.3) +
geom_line() +
geom_point(data = loyn, aes(y = ABUND, x = AREA)) +
scale_y_log10() +
scale_x_log10() +
theme_bw()
loyn.list <- with(loyn, list(AREA = modelr::seq_range(AREA, n=100)))
newdata <- emmeans(loyn.brm3, ~AREA|fGRAZE, at = loyn.list, type='response') |>
as.data.frame()
head(newdata) AREA fGRAZE response lower.HPD upper.HPD
0.10000 1 13.35678 8.347263 21.02803
17.98788 1 23.72537 18.959959 28.43506
35.87576 1 25.60432 21.472996 30.50046
53.76364 1 26.77315 22.353425 31.31426
71.65152 1 27.61484 23.291129 32.27396
89.53939 1 28.27732 23.650035 32.87856
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 ggplot(newdata, aes(y = response, x = AREA)) +
geom_ribbon(aes(ymin = lower.HPD, ymax = upper.HPD, fill = fGRAZE), alpha = 0.3) +
geom_line(aes(color = fGRAZE)) +
theme_bw() +
scale_x_log10() +
scale_y_log10()
spaghetti = emmeans(loyn.brm3, ~AREA|fGRAZE, at = loyn.list, type = 'response') |>
gather_emmeans_draws() |> mutate(Fit = exp(.value))
wch <- sample(1:max(spaghetti$.draw), 100,replace = FALSE)
spaghetti <- spaghetti |> filter(.draw %in% wch)
ggplot(newdata) +
geom_line(data = spaghetti, aes(y = Fit, x = AREA, color = fGRAZE,
group = interaction(fGRAZE,.draw)), alpha = 0.1) +
geom_line(aes(y = response, x = AREA, color = fGRAZE)) +
theme_bw() +
scale_x_log10() + scale_y_log10()
# or honouring the data range
loyn.nd <- loyn |>
group_by(fGRAZE) |>
expand(AREA = modelr::seq_range(AREA, n=100),
DIST = mean(DIST), LDIST = mean(LDIST), ALT = mean(ALT), YR.ISOL = mean(YR.ISOL))
loyn.brm3 |>
epred_draws(newdata = loyn.nd, value = '.value') |>
median_hdci() |>
ggplot(aes(x = AREA, y = .value, colour = fGRAZE, fill = fGRAZE)) +
geom_ribbon(aes(ymin = .lower, ymax = .upper), colour = NA, alpha = 0.3) +
geom_line() +
geom_point(data = loyn, aes(y = ABUND, x = AREA)) +
scale_y_log10() +
scale_x_log10() +
theme_bw()
Lets explore the relationship between bird abundance and patch area for each grazing intensity separately.
loyn.list <- with(loyn, list(AREA = modelr::seq_range(AREA, n = 100)))
newdata <- emmeans(loyn.brm4b, ~AREA|fGRAZE, at = loyn.list, type='response') |>
as.data.frame()
head(newdata) AREA fGRAZE response lower.HPD upper.HPD
0.10000 1 19.47290 12.51028 28.00172
17.98788 1 26.29237 22.78493 30.10815
35.87576 1 27.39322 24.28281 30.72652
53.76364 1 28.04763 24.98364 31.22063
71.65152 1 28.51268 25.33922 31.45183
89.53939 1 28.88930 25.63054 31.82012
Point estimate displayed: median
Results are back-transformed from the log scale
HPD interval probability: 0.95 ggplot(newdata, aes(y = response, x = AREA)) +
geom_ribbon(aes(ymin = lower.HPD, ymax = upper.HPD, fill = fGRAZE), alpha = 0.3) +
geom_line(aes(color = fGRAZE)) +
theme_bw() +
scale_x_log10() +
scale_y_log10()
spaghetti = emmeans(loyn.brm4b, ~AREA|fGRAZE, at = loyn.list, type = 'response') |>
gather_emmeans_draws() |> mutate(Fit = exp(.value))
wch <- sample(1:max(spaghetti$.draw), 100,replace = FALSE)
spaghetti <- spaghetti |> filter(.draw %in% wch)
ggplot(newdata) +
geom_line(data = spaghetti, aes(y = Fit, x = AREA, color = fGRAZE,
group = interaction(fGRAZE,.draw)), alpha = 0.1) +
geom_line(aes(y = response, x = AREA, color = fGRAZE)) +
theme_bw() +
scale_x_log10() + scale_y_log10()
## or honouring the data range
loyn.nd <- loyn |>
group_by(fGRAZE) |>
expand(AREA = modelr::seq_range(AREA, n=100))
loyn.brm4b |>
epred_draws(newdata = loyn.nd, value = '.value') |>
median_hdci() |>
ggplot(aes(x = AREA, y = .value, colour = fGRAZE, fill = fGRAZE)) +
geom_ribbon(aes(ymin = .lower, ymax = .upper), colour = NA, alpha = 0.3) +
geom_line() +
geom_point(data = loyn, aes(y = ABUND, x = AREA)) +
scale_y_log10() +
scale_x_log10() +
theme_bw()
12 References
Loyn, R. H. 1987. “Nature Conservation: The Role of Remnants of Native Vegetation.” In, edited by D. A. Saunders, G. W. Arnold, A. A. Burbridge, and A. J. M. Hopkins. Chipping Norton, NSW: Surrey Beatty & Sons.